What would be the main importance you see on the study of Artinian/Noetherian Modules?
The Noetherian condition makes many results much easier to prove (or in fact a necesary condition!) especially if the ring over which the module is defined is Noetherian. Perhaps the single most important technical device is the following lemma:
Let M be a Noetherian module over a Noetherian ring R, then M is of finite presentation, i.e. there is an exact sequence:
R^m --> R^n --> M --> 0
in other words the module is given by a finite number of generators AND a finite number of relations.
Surprisingly, Artinian modules are much more specialised. They tend to occur when working locally at a prime ideal p (i.e. the module is killed by some power of p, often by starting with a module M and modding out p^n M giving a filtration and exact sequences
0--> p^(k +1) M --> p^k M --> p^k M / p^(k-1) M -> 0
where the latter is a vector space over R_p/pR_p ). The Artinian module is then a stepping stone for working over the completion of R_p at p,
It seems to me that such modules are a natural generalization of a finite dimensional linear space over a field. See, for instance, http://www.springer.com/us/book/9783540353157
The Noetherian condition makes many results much easier to prove (or in fact a necesary condition!) especially if the ring over which the module is defined is Noetherian. Perhaps the single most important technical device is the following lemma:
Let M be a Noetherian module over a Noetherian ring R, then M is of finite presentation, i.e. there is an exact sequence:
R^m --> R^n --> M --> 0
in other words the module is given by a finite number of generators AND a finite number of relations.
Surprisingly, Artinian modules are much more specialised. They tend to occur when working locally at a prime ideal p (i.e. the module is killed by some power of p, often by starting with a module M and modding out p^n M giving a filtration and exact sequences
0--> p^(k +1) M --> p^k M --> p^k M / p^(k-1) M -> 0
where the latter is a vector space over R_p/pR_p ). The Artinian module is then a stepping stone for working over the completion of R_p at p,
Hello,
Consider the algebraic polynomials in a finite number of variables, which is a Noetherian ring. Then any ideal is represented by a finite number of polynomials (Hilbert basis theorem), although the dimension of this ring as a K-vector space is infinite.
For rings of differential polynomials, there are some similar results on radical differential ideals (Ritt-Raudenbusch theorem) : they are in a sense Noetherian.
I have no example for the importance of Artinian rings, I guess you should find something in the Number Theory.
Besides all these algebraic reasons are the geometrical: the study
of k-algebras finitely generated (they are Noetherian because of
Hilbert's Basis) is the algebraic counter part of classical
algebraic geometry. In fact, they can realized as the ring of
regular functions of affine varieties. Almost every theorem on
Noetherian rings have a geometric interpretation. For instance,
given any ideal in a Noetherian ring there a finite number of
primes ideal minimal among those containing the ideal,
geometrically this means that the variety associated with the
ideal has a finite number of irreducible components. In this
context, artinian rings also have a clear interpretation: they are
the ring of functions of algebraic finite sets.
Geometrically, studying finitely generated modules on noetherian
rings corresponds to the study of vector bundles on the associated
variety, or more generally, to the study of coherent sheafs. You
can take a look at Kemper's book on Commutative Algebra for the
basics on this correspondence.
The benefit of studying Artinian/Noetherian Modules (modules with chain conditions) which appears to me is connected with the decomposition of a module into a finite number of indecompsable submodules which is the first step towards analysing the structure of modules as well as classifying them. Such decomposition exists if a module satisfies either chain condition on submodules.
Every subspace of finite dimensional vector space is finite-dimensional and the dimension of a subspace is at most the dimension of the whole space. The naive analogue of this for modules and submodules is wrong: For example a submodule of a finitely generated module need not be finitely generated. Artinian/Noetherian methods may be use for solve this problems.
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